If the consumer had 100 units of x and 400 units of y, what would be the consumer's marginal rate of substitution between the two goods (using either definition from the seminar slides, but not the definition from the textbook)?

Question

BlackTom AI · Accepted Answer

The task presents a numerical question about the marginal rate of substitution (MRS) but does not provide any answer options to analyze. To work through this, first restate what is being asked: determining the consumer's marginal rate of substitution between goods x and y given the consumer has 100 units of x and 400 units of y, using definitions from the seminar slides rather than a textbook definition.

Key point: The MRS is not determined by the quantities alone in isolation. It depends on the consumer's underlying utility function or, at minimum, on the definition used for MRS (how MU_x and MU_y are defined and how they relate). Without a specified utility function or a specified form of marginal utilities, you cannot compute a numerical MRS from the endowment alone.

What you would need to compute MRS_xy (the rate of substitution of x for y):
- A defined utility function u(x, y), or
- Explicit marginal utilities MU_x and MU_y as given in the seminar slides, or
- The definition/formula for MRS being used in the seminar (for example, MRS_xy = MU_x / MU_y, or the alternative sign convention MRS_yx = MU_y / MU_x, along with the domain conventions).

If you had a common form for the utility function, you could proceed as follows:
- From u(x, y), compute MU_x = ∂u/∂x and MU_y = ∂u/∂y.
- Then MRS_xy = MU_x / MU_y, representing how many units of y the consumer is willing to give up for an extra unit of x, holding utility constant.

Illustrative (but non-exhaustive) scenarios for what could yield a numerical MRS:
- If the utility function is a Cobb-Douglas u(x, y) = x^a y^(1−a), then MU_x = a x^(a−1) y^(1−a) and MU_y = (1−a) x^a y^(−a); thus MRS_xy = [a/(1−a)] * (y/x).
- If the utility is perfect complements, u(x, y) = min{x, y}, the MRS is not well-defined at all points (it jumps depending on the region).
- If the utility is a linear functionality like u(x, y) = αx + βy, MU_x = α and MU_y = β, so MRS_xy = α/β, a constant determined by α and β, not by the endowment.

Because the question provides only endowment values (100 x, 400 y) and no utility specification or MRS definition from the slides, you cannot compute a numeric MRS from this information alone. If you can supply the seminar’s MRS definition or the utility function used in the course materials, you could plug in the derivatives and obtain a concrete value.

Summary of potential issues with missing information:
- No answer options to evaluate for correctness or incorrectness.
- No utility function or marginal utility expressions supplied to compute MU_x and MU_y.
- Endowment amounts by themselves do not determine MRS without preferences/usages specified in the material from which the MRS is defined.

ECC2000 - ECC5900 - S2 2025 ECC2000/5900 Mid-Semester Test - Quiz - NO SAFE EXAM BROWSWER NEEDED - 8 September 2025, opens 2pm (Melbourne, Australian time) duplicate

If the consumer had 100 units of x and 400 units of y, what would be the consumer's marginal rate of substitution between the two goods (using either definition from the seminar slides, but not the definition from the textbook)?

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